Theorem gsumhashmul 31325

Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)

Hypotheses

Ref	Expression
gsumhashmul.b	⊢ 𝐵 = (Base‘𝐺)
gsumhashmul.z	⊢ 0 = (0g‘𝐺)
gsumhashmul.x	⊢ · = (.g‘𝐺)
gsumhashmul.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumhashmul.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumhashmul.1	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsumhashmul	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))

Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥

Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul

Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumhashmul.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		suppssdm 8002	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
3	2, 1	fssdm 6629	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
4	1, 3	feqresmpt 6847	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥)))
5	4	oveq2d 7300	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))))
6		gsumhashmul.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
7		gsumhashmul.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
8		gsumhashmul.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
9		gsumhashmul.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp 0 )
10		relfsupp 9139	. . . . . . . . 9 ⊢ Rel finSupp
11	10	brrelex1i 5644	. . . . . . . 8 ⊢ (𝐹 finSupp 0 → 𝐹 ∈ V)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
13	1	ffnd 6610	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14	12, 13	fndmexd 7762	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
15		ssidd 3945	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
16	6, 7, 8, 14, 1, 15, 9	gsumres 19523	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(1st ‘𝑧))
18		fveq2 6783	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑧) → (𝐹‘𝑥) = (𝐹‘(1st ‘𝑧)))
19	9	fsuppimpd 9144	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20		ssidd 3945	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
21	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴⟶𝐵)
22	3	sselda 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ 𝐴)
23	21, 22	ffvelrnd 6971	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹‘𝑥) ∈ 𝐵)
24	1	ffund 6613	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
25		funrel 6458	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
26		reldif 5727	. . . . . . . . 9 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
27	24, 25, 26	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28		1stdm 7890	. . . . . . . 8 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
29	27, 28	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
30	7	fvexi 6797	. . . . . . . . . . . 12 ⊢ 0 ∈ V
31	30	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ V)
32		fressupp 31031	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
33	24, 12, 31, 32	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
34	33	dmeqd 5817	. . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
35	2	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36		ssdmres 5917	. . . . . . . . . 10 ⊢ ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
37	35, 36	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
38	34, 37	eqtr3d 2781	. . . . . . . 8 ⊢ (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
39	38	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
40	29, 39	eleqtrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ (𝐹 supp 0 ))
41	24	funresd 6484	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
42	41	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
43	37	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
44	43	biimpar 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
46	45	fvresd 6803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
47		funopfvb 6834	. . . . . . . . . . 11 ⊢ ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
48	47	biimpa 477	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥)) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
49	42, 44, 46, 48	syl21anc 835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
50	33	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
51	49, 50	eleqtrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52		eqeq2 2751	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → (𝑧 = 𝑣 ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
53	52	bibi2d 343	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → ((𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
54	53	ralbidv 3113	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	54	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
56		fvexd 6798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ V)
57	27	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59		1st2nd 7889	. . . . . . . . . . . . . . . . 17 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
60	57, 58, 59	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
61		opeq1 4805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ⟨𝑥, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
62	61	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ⟨𝑥, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
63	60, 62	eqtr4d 2782	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩)
64		difssd 4068	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
65	64	sselda 3922	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹)
66	65	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ 𝐹)
67	63, 66	eqeltrrd 2841	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹)
68	63, 67	jca 512	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩ ∧ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹))
69		opeq2 4806	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd ‘𝑧)⟩)
70	69	eqeq2d 2750	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩))
71	69	eleq1d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹))
72	70, 71	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩ ∧ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹)))
73	56, 68, 72	spcedv 3538	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74		vex 3437	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
75	74	elsnres 5934	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
76	73, 75	sylibr 233	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
77	13	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝐹 Fn 𝐴)
78	22	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑥 ∈ 𝐴)
79		fnressn 7039	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
80	77, 78, 79	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
81	76, 80	eleqtrd 2842	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹‘𝑥)⟩})
82		elsni 4579	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {⟨𝑥, (𝐹‘𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
83	81, 82	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
84		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
85	84	fveq2d 6787	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → (1st ‘𝑧) = (1st ‘⟨𝑥, (𝐹‘𝑥)⟩))
86		fvex 6796	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
87	74, 86	op1st 7848	. . . . . . . . . . 11 ⊢ (1st ‘⟨𝑥, (𝐹‘𝑥)⟩) = 𝑥
88	85, 87	eqtr2di 2796	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → 𝑥 = (1st ‘𝑧))
89	83, 88	impbida 798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
90	89	ralrimiva 3104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
91	51, 55, 90	rspcedvd 3564	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣))
92		reu6 3662	. . . . . . 7 ⊢ (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st ‘𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣))
93	91, 92	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st ‘𝑧))
94	17, 6, 7, 18, 8, 19, 20, 23, 40, 93	gsummptf1o 19573	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))))
95	5, 16, 94	3eqtr3d 2787	. . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))))
96		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
97	96	eldifad 3900	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹)
98		funfv1st2nd 7896	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝐹‘(1st ‘𝑧)) = (2nd ‘𝑧))
99	24, 97, 98	syl2an2r 682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st ‘𝑧)) = (2nd ‘𝑧))
100	99	mpteq2dva 5175	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧)))
101	100	oveq2d 7300	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))))
102	95, 101	eqtrd 2779	. . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))))
103		nfcv 2908	. . . 4 ⊢ Ⅎ𝑧(1st ‘𝑡)
104		fvex 6796	. . . . 5 ⊢ (2nd ‘𝑡) ∈ V
105		fvex 6796	. . . . 5 ⊢ (1st ‘𝑡) ∈ V
106	104, 105	op2ndd 7851	. . . 4 ⊢ (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ → (2nd ‘𝑧) = (1st ‘𝑡))
107		resfnfinfin 9108	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
108	13, 19, 107	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
109	33, 108	eqeltrrd 2841	. . . 4 ⊢ (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
110	33	rneqd 5850	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111		rnresss 5930	. . . . . 6 ⊢ ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
112	1	frnd 6617	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
113	111, 112	sstrid 3933	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114	110, 113	eqsstrrd 3961	. . . 4 ⊢ (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115		2ndrn 7891	. . . . 5 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd ‘𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
116	27, 115	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd ‘𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117		relcnv 6015	. . . . . . . 8 ⊢ Rel ◡𝐹
118		reldif 5727	. . . . . . . 8 ⊢ (Rel ◡𝐹 → Rel (◡𝐹 ∖ ({ 0 } × V)))
119	117, 118	mp1i 13	. . . . . . 7 ⊢ (𝜑 → Rel (◡𝐹 ∖ ({ 0 } × V)))
120		1st2nd 7889	. . . . . . 7 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
121	119, 120	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
122		cnvdif 6052	. . . . . . . . . 10 ⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ◡(V × { 0 }))
123		cnvxp 6065	. . . . . . . . . . 11 ⊢ ◡(V × { 0 }) = ({ 0 } × V)
124	123	difeq2i 4055	. . . . . . . . . 10 ⊢ (◡𝐹 ∖ ◡(V × { 0 })) = (◡𝐹 ∖ ({ 0 } × V))
125	122, 124	eqtri 2767	. . . . . . . . 9 ⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ({ 0 } × V))
126	125	eqimss2i 3981	. . . . . . . 8 ⊢ (◡𝐹 ∖ ({ 0 } × V)) ⊆ ◡(𝐹 ∖ (V × { 0 }))
127	126	a1i 11	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆ ◡(𝐹 ∖ (V × { 0 })))
128	127	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 ∈ ◡(𝐹 ∖ (V × { 0 })))
129	121, 128	eqeltrrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ ◡(𝐹 ∖ (V × { 0 })))
130	105, 104	opelcnv 5793	. . . . 5 ⊢ (⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ ◡(𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131	129, 130	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
132	27	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133		eqidd 2740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} = ∪ ◡{𝑧})
134		cnvf1olem 7959	. . . . . . . . 9 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ ∪ ◡{𝑧} = ∪ ◡{𝑧})) → (∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪ ◡{∪ ◡{𝑧}}))
135	134	simpld 495	. . . . . . . 8 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ ∪ ◡{𝑧} = ∪ ◡{𝑧})) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })))
136	132, 96, 133, 135	syl12anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })))
137	136, 125	eleqtrdi 2850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ (◡𝐹 ∖ ({ 0 } × V)))
138		eqeq2 2751	. . . . . . . . 9 ⊢ (𝑢 = ∪ ◡{𝑧} → (𝑡 = 𝑢 ↔ 𝑡 = ∪ ◡{𝑧}))
139	138	bibi2d 343	. . . . . . . 8 ⊢ (𝑢 = ∪ ◡{𝑧} → ((𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
140	139	ralbidv 3113	. . . . . . 7 ⊢ (𝑢 = ∪ ◡{𝑧} → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
141	140	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = ∪ ◡{𝑧}) → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
142	117, 118	mp1i 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → Rel (◡𝐹 ∖ ({ 0 } × V)))
143		simplr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)))
144		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
145		df-rel 5597	. . . . . . . . . . . . . 14 ⊢ (Rel (◡𝐹 ∖ ({ 0 } × V)) ↔ (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146	119, 145	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147	146	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148	147, 143	sseldd 3923	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 ∈ (V × V))
149		2nd1st 7888	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (V × V) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
150	148, 149	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
151	144, 150	eqtr4d 2782	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑧 = ∪ ◡{𝑡})
152		cnvf1olem 7959	. . . . . . . . . 10 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪ ◡{𝑡})) → (𝑧 ∈ ◡(◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = ∪ ◡{𝑧}))
153	152	simprd 496	. . . . . . . . 9 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪ ◡{𝑡})) → 𝑡 = ∪ ◡{𝑧})
154	142, 143, 151, 153	syl12anc 834	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 = ∪ ◡{𝑧})
155	27	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
156	96	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 = ∪ ◡{𝑧})
158		cnvf1olem 7959	. . . . . . . . . . 11 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = ∪ ◡{𝑧})) → (𝑡 ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪ ◡{𝑡}))
159	158	simprd 496	. . . . . . . . . 10 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = ∪ ◡{𝑧})) → 𝑧 = ∪ ◡{𝑡})
160	155, 156, 157, 159	syl12anc 834	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 = ∪ ◡{𝑡})
161	146	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162		simplr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)))
163	161, 162	sseldd 3923	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 ∈ (V × V))
164	163, 149	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
165	160, 164	eqtrd 2779	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
166	154, 165	impbida 798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧}))
167	166	ralrimiva 3104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧}))
168	137, 141, 167	rspcedvd 3564	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢))
169		reu6 3662	. . . . 5 ⊢ (∃!𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ ∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢))
170	168, 169	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
171	103, 6, 7, 106, 8, 109, 114, 116, 131, 170	gsummptf1o 19573	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))) = (𝐺 Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡))))
172		fveq2 6783	. . . . . 6 ⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧))
173	172	cbvmptv 5188	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡)) = (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑧))
174	33	cnveqd 5787	. . . . . . 7 ⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) = ◡(𝐹 ∖ (V × { 0 })))
175	174, 125	eqtr2di 2796	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) = ◡(𝐹 ↾ (𝐹 supp 0 )))
176	175	mpteq1d 5170	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑧)) = (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧)))
177	173, 176	eqtrid 2791	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡)) = (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧)))
178	177	oveq2d 7300	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡))) = (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))))
179	102, 171, 178	3eqtrd 2783	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))))
180		nfcv 2908	. . 3 ⊢ Ⅎ𝑦(1st ‘𝑧)
181		nfv 1918	. . 3 ⊢ Ⅎ𝑥𝜑
182		vex 3437	. . . 4 ⊢ 𝑦 ∈ V
183	74, 182	op1std 7850	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
184		relcnv 6015	. . . 4 ⊢ Rel ◡(𝐹 ↾ (𝐹 supp 0 ))
185	184	a1i 11	. . 3 ⊢ (𝜑 → Rel ◡(𝐹 ↾ (𝐹 supp 0 )))
186		cnvfi 8972	. . . 4 ⊢ ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187	108, 186	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188	112	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹 ⊆ 𝐵)
189	184	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → Rel ◡(𝐹 ↾ (𝐹 supp 0 )))
190		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )))
191		1stdm 7890	. . . . . . 7 ⊢ ((Rel ◡(𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )))
192	189, 190, 191	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )))
193		df-rn 5601	. . . . . 6 ⊢ ran (𝐹 ↾ (𝐹 supp 0 )) = dom ◡(𝐹 ↾ (𝐹 supp 0 ))
194	192, 193	eleqtrrdi 2851	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195	111, 194	sselid 3920	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ ran 𝐹)
196	188, 195	sseldd 3923	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ 𝐵)
197	180, 181, 6, 183, 185, 187, 8, 196	gsummpt2d 31318	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))) = (𝐺 Σg (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198		df-ima 5603	. . . . . . 7 ⊢ (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199		supppreima 31034	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
200	24, 12, 31, 199	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
201	200	imaeq2d 5972	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))))
202	198, 201	eqtr3id 2793	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))))
203		funimacnv 6522	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
204	24, 203	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205		difssd 4068	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206		df-ss 3905	. . . . . . 7 ⊢ ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207	205, 206	sylib 217	. . . . . 6 ⊢ (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208	202, 204, 207	3eqtrd 2783	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209	193, 208	eqtr3id 2793	. . . 4 ⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
210	8	cmnmndd 19418	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
211	210	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212	108	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213		imafi2 31055	. . . . . . 7 ⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214	212, 186, 213	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215	193, 113	eqsstrrid 3971	. . . . . . 7 ⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216	215	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ 𝐵)
217		gsumhashmul.x	. . . . . . 7 ⊢ · = (.g‘𝐺)
218	6, 217	gsumconst 19544	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥 ∈ 𝐵) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219	211, 214, 216, 218	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220		cnvresima 6138	. . . . . . . 8 ⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221	209	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222	221	biimpa 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223	222	snssd 4743	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224		sspreima 6954	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
225	24, 223, 224	syl2an2r 682	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
226	200	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
227	225, 226	sseqtrrd 3963	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228		df-ss 3905	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥}))
229	227, 228	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥}))
230	220, 229	eqtr2id 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) = (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231	230	fveq2d 6787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(◡𝐹 “ {𝑥})) = (♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232	231	oveq1d 7299	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(◡𝐹 “ {𝑥})) · 𝑥) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233	219, 232	eqtr4d 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))
234	209, 233	mpteq12dva 5164	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥)))
235	234	oveq2d 7300	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))
236	179, 197, 235	3eqtrd 2783	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2107  ∀wral 3065  ∃wrex 3066  ∃!wreu 3067  Vcvv 3433   ∖ cdif 3885   ∩ cin 3887   ⊆ wss 3888  {csn 4562  ⟨cop 4568  ∪ cuni 4840   class class class wbr 5075   ↦ cmpt 5158   × cxp 5588  ^◡ccnv 5589  dom cdm 5590  ran crn 5591   ↾ cres 5592   “ cima 5593  Rel wrel 5595  Fun wfun 6431   Fn wfn 6432  ⟶wf 6433  ‘cfv 6437  (class class class)co 7284  1^st c1st 7838  2^nd c2nd 7839   supp csupp 7986  Fincfn 8742   finSupp cfsupp 9137  ♯chash 14053  Basecbs 16921  0_gc0g 17159   Σ_g cgsu 17160  Mndcmnd 18394  ._gcmg 18709  CMndccmn 19395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-of 7542  df-om 7722  df-1st 7840  df-2nd 7841  df-supp 7987  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-fsupp 9138  df-oi 9278  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-nn 11983  df-2 12045  df-n0 12243  df-z 12329  df-uz 12592  df-fz 13249  df-fzo 13392  df-seq 13731  df-hash 14054  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-0g 17161  df-gsum 17162  df-mre 17304  df-mrc 17305  df-acs 17307  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-submnd 18440  df-mulg 18710  df-cntz 18932  df-cmn 19397

This theorem is referenced by:  elrspunidl  31615